It has been a long time since I posted any new items. In a caravan with no electricity and no mobile phone reception there is no way of having my own computer, not even with wifi.

In the meantime the Australian Museum in Sydney was kind enough to show a display of mine for 2 weeks as part of the Alive! Exhibition on biodiversity, in September.

Summer is coming and the wild oats are up to 8 ft tall. I am having trouble looking after 4 acres of land by myself working only with hand tools. Pink and red bush roses are abundantly in flower after the unusual rains of winter and spring. Chocolate lilies ( yes they do have this perfume) are flowering in grassy woodland remnants.

Mindful of last year's devastating bushfires I am publishing the equations I have found in connection with the Fibonacci Series. If my caravan were to burn down then all my papers and models would be lost. I am only briefly noting them. There is a lot to document and it might be a slow process. So here goes:-

1.

F30 = F29 + F28=317811 + 196418 = 514229

2.

F17 = 2F15 + F14 = 2x377 + 233 = 754 + 233 = 987

F20 = 2F18 + F17 = 2x1597 + 987 = 3194 + 987 = 4181

F30 = 2F28 + F27 = 2x196418 + 121393 = 514229

3.

F1 +....F10 = 0+1+1+2+3+5+8+13+21+34 = 88 = 89-1

F1 + F2 + ....F16 = F18 - 1 = 0+1+1+2+3+....+610 = 1596 = 1597-1

4.

F14 = 7x F10 - F6 = (7 x 34) - 5 = 238 - 5 = 233 = F14

5.

6.

7.

........................................................................................

F8/F7 = 13/8 = 1.625

F10/F9 = 34/21 = 1.6190476

F12/F11 = 89/55 = 1.6181818

F16/F15 = 610/377 = 1.6180371

F18/F17 = 1597/987 = 1.6180344

F20/F19 = 4181/2584 = 1.618034

F30/F29 = 514229/317811 = 1.6180339

F40/F39 = 63245986/39088169 = 1.6180339

In the meantime the Australian Museum in Sydney was kind enough to show a display of mine for 2 weeks as part of the Alive! Exhibition on biodiversity, in September.

Summer is coming and the wild oats are up to 8 ft tall. I am having trouble looking after 4 acres of land by myself working only with hand tools. Pink and red bush roses are abundantly in flower after the unusual rains of winter and spring. Chocolate lilies ( yes they do have this perfume) are flowering in grassy woodland remnants.

Mindful of last year's devastating bushfires I am publishing the equations I have found in connection with the Fibonacci Series. If my caravan were to burn down then all my papers and models would be lost. I am only briefly noting them. There is a lot to document and it might be a slow process. So here goes:-

1.

**F(n) = F(n-1) + F(n-2)....**this is well know; it is definition of the seriesF30 = F29 + F28=317811 + 196418 = 514229

2.

**F(n) = 2F(n-2) + F(n-3).**......this is used for seashell spiralsF17 = 2F15 + F14 = 2x377 + 233 = 754 + 233 = 987

F20 = 2F18 + F17 = 2x1597 + 987 = 3194 + 987 = 4181

F30 = 2F28 + F27 = 2x196418 + 121393 = 514229

3.

**F(1) + F(2) + F(3) +.....+F(n) = F(n + 2) - 1**....this is sum totalF1 +....F10 = 0+1+1+2+3+5+8+13+21+34 = 88 = 89-1

F1 + F2 + ....F16 = F18 - 1 = 0+1+1+2+3+....+610 = 1596 = 1597-1

4.

**F(n) = 7F(n-4) - F(n-8)....**this came up while working out how to crochet a whirlpoolF14 = 7x F10 - F6 = (7 x 34) - 5 = 238 - 5 = 233 = F14

5.

**F(n) = 2F(n-4) + 3F(n-3) ....**further "unpacking" of the series.6.

**F(n) = 5F(n-4) + 3F(n-5)....**ditto of above7.

**F(n) = 5F(n-6) + 8F(n-5)....**ditto........................................................................................

**8. The following are spookily exact especially at the higher numbers, not always so at lowest iterations:-**

**F(n) / F(n-1)**F8/F7 = 13/8 = 1.625

F10/F9 = 34/21 = 1.6190476

F12/F11 = 89/55 = 1.6181818

F16/F15 = 610/377 = 1.6180371

F18/F17 = 1597/987 = 1.6180344

F20/F19 = 4181/2584 = 1.618034

F30/F29 = 514229/317811 = 1.6180339

F40/F39 = 63245986/39088169 = 1.6180339

..................................................................................

****This is Phi, a "transcedental number

**",**1.6180339887498

The golden mean is 1:618

or 2:3:5

****There is

**a most extraordinary website**http://www.humanresonance.org/

The author Alexander Putney has written a book on Phi. One can read it online for free.

NB. 5/2013 Incredibly, this website has been stolen! The author had to buy a new web domain.

It is http://www.human-resonance.org

...................................................................................

**F(n)/F(n-2)**

F12/F10 = 89/34 = 2.617647

F20/F18 = 4181/1597 = 2.6180338

F40/F38 = 63245986/24157817 = 2.6180339

**F(n)/F(n-3)**

F12/F9 = 89/21 = 4.2380952

F20/F17 = 4181/ 987 = 4.2360688

F40/F37 = 63245986/14930352 = 4.2360679

**F(n)/F(n-4)**

F12/F8 = 89/13 = 6.8461538

F20/F16 = 4181/610 = 6.8540983

F40/F36 = 63245986/9227465 = 6.8541019

**F(n)/F(n-5)**

F20/F15 = 4181/377 = 11.090185

F36/F31 = 9227465/832040 = 11.090169

F40/F35 = 63245986/5702887 = 11.090169

**F(n)/F(n-6)**

F16/F10 = 610/34 = 17.941176

F26/F20 = 75025/4181 = 17.944271

F40/F34 = 63245986/3524578 = 17.944271

**F(n)/F(n-7)**

F10/F3 = 34/1 = 34

F11/F4 = 55/2 = 27.5

F17/F10 = 987/34 = 29.029411

F33/F26 = 2178309/75025 = 29.034441

F40/F33 = 63245986/2178309 = 29.034441

**F(n)/F(n-10)**

F26/F16 = 75025/610 = 122.9918

F36/F26 = 9227465/75025 = 122.99186

F40/F30 = 63245986/514229 = 122.99186

**....................................................**

**F(n)/F(n-20)**

F27/F7 = 121393/8 = 15174.125

F30/F10 = 514229/34 = 15124.382

F55/F35 = 86267571272/5702887

divide top and bottom each by 1000, ie 1.

=86267571/5702.887 = 15126.999

F60/F40 =

956722026041/63245986 = 95672203/6324.5986

956722026041/63245986 = 95672203/6324.5986

= 15126.999

F90/F70 = 1779978866004714/117669030460994

= 17799789/1176.69 = 15126.9999

Amazing, isn't it, and elegant?

It is fascinating playing around with these numbers. There is intricate correlation indeed!

Cheerio from Tiiu Vanamois till later.

9. Half F(n) = 9[ 0.5 x 2F(n-8) + F(n-9)] - half F(n-12)

........ for knitting a half clam shell with wavy lips!

10. Half F(n) = 9[ 0.5x2F(n-7) + F(n-8)] - half F(n-13)

for half clam

11. There is also Half F(n) = 9[F(n-6)] - half F(n-12)

These lines got placed oddly.

One day I might be able to elaborate more fully

but I think it is beyond the scope of a little blog.

Cheerio from Tiiu Vanamois till later.

9. Half F(n) = 9[ 0.5 x 2F(n-8) + F(n-9)] - half F(n-12)

........ for knitting a half clam shell with wavy lips!

10. Half F(n) = 9[ 0.5x2F(n-7) + F(n-8)] - half F(n-13)

for half clam

11. There is also Half F(n) = 9[F(n-6)] - half F(n-12)

**0.5 x 2Fn denotes knit 2 together to make 1.**

These equations are not quite right yet but they are on the way.These equations are not quite right yet but they are on the way

These lines got placed oddly.

One day I might be able to elaborate more fully

but I think it is beyond the scope of a little blog.